# First-Order ODE Solver (Separation of Variables) Solve separable dy/dx = g(x)·h(y) symbolically: H(y) = G(x) + C, explicit inversion, initial conditions, and partial fractions for quadratics like y(1−y). > Canonical page: https://elysiatools.com/en/tools/first-order-ode-solver - **Category:** Math & Numbers - **Keywords:** first order ode, separable differential equation, separation of variables, dy/dx, initial value problem, partial fractions, logistic equation, implicit solution, calculus, symbolic integration ## Overview Solve separable first-order ordinary differential equations of the form dy/dx = g(x)·h(y) symbolically with step-by-step integration, algebraic inversion for explicit solutions, and exact constant evaluation for initial value problems. ## Inputs - **g(x) — x-side factor** (text): x-dependent factor of dy/dx = g(x)·h(y). Terms: constants, x^n, (kx+m)^n, e^(kx), sin(kx+m), cos(kx+m), ln(kx+m), sqrt(kx+m). - **h(y) — y-side factor** (text): y-dependent factor. Single terms (y, y^n, (ky+m)^n, e^(ky)) or a degree ≤ 2 polynomial with real roots, or c/D(y). - **x₀ (initial x)** (number): Only used together with y₀ to pin down the constant C. - **y₀ = y(x₀)** (number): Initial value. Leave empty for the general solution; provide with x₀ for the particular solution. ## When to use - Solving first-order separable differential equations analytically for calculus and engineering coursework. - Evaluating initial value problems (IVPs) to determine the exact integration constant C from given coordinates (x₀, y₀). - Integrating rational and quadratic y-expressions, such as logistic equations, using symbolic partial fraction decomposition. ## How it works - Enter the x-dependent factor g(x) and y-dependent factor h(y) that define the separable equation dy/dx = g(x)·h(y). - Optionally enter initial condition values x₀ and y₀ to solve for the particular integration constant C. - The solver rewrites the equation as (1/h(y)) dy = g(x) dx, integrates both sides symbolically, and attempts to isolate y for an explicit solution. ## Use cases - Verifying step-by-step symbolic solutions and integration steps for differential calculus assignments. - Determining explicit trajectories for initial value problems in physics and rate-of-change modeling. - Deriving analytical general and particular solutions for logistic growth equations in biology and economics. ## Frequently asked questions ### What differential equation formats are supported? The solver handles first-order separable equations structured as dy/dx = g(x)·h(y). ### How are quadratic y-terms like y(1−y) integrated? The tool uses partial fraction decomposition on 1/h(y) before integrating each logarithmic term. ### How do I calculate a particular solution instead of a general solution? Provide numerical values for both x₀ and y₀ to evaluate the constant C. ### What happens if y cannot be algebraically isolated? The tool outputs the exact implicit equation H(y) = G(x) + C. ### Which mathematical expressions can be entered for g(x)? 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